Dynamics of SL2(R) over moduli space in genus two Curtis T. McMullen∗ 19 October, 2003 Abstract This paper classifies orbit closures and invariant measures for the nat- ural action of SL2(R) on ΩM2, the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two. Contents 1 Introduction.............................. 1 2 DynamicsandLiegroups ...................... 6 3 Riemannsurfacesandholomorphic1-forms . 12 4 Abelian varieties with real multiplication . .. 15 5 Recognizingeigenforms . .. .. .. .. .. .. .. .. .. 22 6 Algebraicsumsof1-forms .. .. .. .. .. .. .. .. .. 26 7 Connectedsumsof1-forms . .. .. .. .. .. .. .. .. 29 8 Eigenformsasconnectedsums. 34 9 Pairsofsplittings........................... 36 10 Dynamics on Ω 2(2) ........................ 42 11 Dynamics on ΩM (1, 1)....................... 47 M2 12 Dynamics on ΩED .......................... 51 1 Introduction Let denote the moduli space of Riemann surfaces of genus g. By Teichm¨uller Mg theory, every holomorphic 1-form ω(z) dz on a surface X g generates a 2 ∈ M complex geodesic f : H g, isometrically immersed for the Teichm¨uller metric. → M In this paper we will show: 2 Theorem 1.1 Let f : H 2 be a complex geodesic generated by a holomor- phic 1-form. Then f(H2) is→M either an isometrically immersed algebraic curve, a Hilbert modular surface, or the full space . M2 ∗Research partially supported by the NSF. 1 In particular, f(H2) is always an algebraic subvariety of . M2 Raghunathan’s conjectures. For comparison, consider a finite volume hy- perbolic manifold M in place of g. While the closure of a geodesicM line in M can be rather wild, the closure of a geodesic plane f : H2 M = Hn/Γ → is always an immersed submanifold. Indeed, the image of f can be lifted to R an orbit of U = SL2( ) on the frame bundle FM ∼= G/Γ, G = SO(n, 1). Raghunathan’s conjectures, proved by Ratner, then imply that Ux = Hx G/Γ ⊂ for some closed subgroup H G meeting xΓx−1 in a lattice. Projecting back to M one finds that f(H2) ⊂ M is an immersed hyperbolic k-manifold with 2 k n [Sh]. ⊂ ≤ ≤ The study of complex geodesics in g is similarly related to the dynamics of SL (R) on the bundle of holomorphicM 1-forms Ω . 2 Mg →Mg A point (X,ω) Ω g consists of a compact Riemann surface of genus g equipped with a holomorphic∈ M 1-form ω Ω(X). The Teichm¨uller geodesic flow, coupled with the rotations ω eiθω∈, generates an action of SL (R) on 7→ 2 Ω g. This action preserves the subspace Ω1 g of unit forms, those satisfying M 2 M X ω = 1. |The| complex geodesic generated by (X,ω) Ω is simply the projection R ∈ 1Mg to g of its SL2(R)-orbit. Our main result is a refinement of Theorem 1.1 whichM classifies these orbits for genus two. Theorem 1.2 Let Z = SL2(R) (X,ω) be an orbit closure in Ω1 2. Then exactly one of the following holds:· M 1. The stabilizer SL(X,ω) of (X,ω) is a lattice, we have Z = SL (R) (X,ω), 2 · and the projection of Z to moduli space is an isometrically immersed Te- ichm¨uller curve V . ⊂M2 2. The Jacobian of X admits real multiplication by a quadratic order of dis- criminant D, with ω as an eigenform, but SL(X,ω) is not a lattice. Then Z =Ω1ED coincides with the space of all eigenforms of discriminant D, and its pro- jection to is a Hilbert modular surface. M2 3. The form ω has a double zero, but is not an eigenform for real multiplica- tion. Then Z =Ω (2) 1M2 coincides with the stratum of all forms with double zeros. It projects sur- jectively to . M2 2 4. The form ω has simple zeros, but is not an eigenform for real multiplica- tion. Then its orbit is dense: we have Z =Ω . 1M2 We note that in case (1) above, ω is also an eigenform (cf. Corollary 5.9). Corollary 1.3 The complex geodesic generated by (X,ω) is dense in 2 iff (X,ω) is not an eigenform for real multiplication. M + Corollary 1.4 Every orbit closure GL2 (R) (X,ω) Ω 2 is a complex orb- ifold, locally defined by linear equations in period· coordin⊂ ates.M Invariant measures. In the setting of Lie groups and homogeneous spaces, it is also known that every U-invariant measure on G/Γ is algebraic (see 2). Similarly, in 10—12 we show: § §§ Theorem 1.5 Each orbit closure Z carries a unique ergodic, SL2(R)-invariant probability measure µZ of full support, and these are all the ergodic probability measures on Ω . 1M2 In terms of local coordinates given by the relative periods of ω, the measure µZ is simply Euclidean measure restricted to the ‘unit sphere’ defined by ω 2 =1 (see 3, 8). | | § § R Pseudo-Anosov mappings. The classification of orbit closures also sheds light on the topology of complexified loops in . M2 Let φ Mod2 ∼= π1( 2) be a pseudo-Anosov element of the mapping class group of∈ a surface of genusM two. Then there is a real Teichm¨uller geodesic γ : R g whose image is a closed loop representing [φ]. Complexifying γ, we obtain→ M a totally geodesic immersion f : H2 →Mg satisfying γ(s)= f(ie2s). The map f descends to the Riemann surface V = H2/Γ , Γ = A Aut(H2) : f(Az)= f(z) . φ φ φ { ∈ } Theorem 1.6 For any pseudo-Anosov element φ π1( 2) with orientable foliations, either ∈ M 1. Γ is a lattice, and f(V ) is a closed algebraic curve, or φ φ ⊂M2 2. Γφ is an infinitely generated group, and f(Vφ) is a Hilbert modular surface. 1 Proof. The limit set of Γφ is the full circle S∞ [Mc2], and f(Vφ) is the projection of the SL2(R)-orbit of an eigenform (by Theorem 5.8 below). Thus we are in case (1) of Theorem 1.2 if Γφ is finitely generated, and otherwise in case (2). 3 In particular, the complexification of a closed geodesic as above is never dense in . Explicit examples where (2) holds are given in [Mc2]. M2 Connected sums. A central role in our approach to dynamics on Ω 2 is played by the following result ( 7): M § Theorem 1.7 Any form (X,ω) of genus two can be written, in infinitely many ways, as a connected sum (X,ω) = (E1,ω1)#(E2,ω2) of forms of genus one. I Here (Ei,ωi) = (C/Λi, dz) are forms in Ω 1, and I = [0, v] is a segment in R2 C M ∼= . The connected sum is defined by slitting each torus Ei open along the image of I in C/Λi, and gluing corresponding edges to obtain X (Figure 1). The forms ωi on Ei combine to give a form ω on X with two zeros at the ends of the slits. We also refer to a connected sum decomposition as a splitting of (X,ω). Figure 1. The connected sum of a pair of tori. Connected sums provide a geometric characterization of eigenforms ( 8): § Theorem 1.8 If (X,ω) Ω 2 has two different splittings with isogenous sum- mands, then it is an eigenform∈ M for real multiplication. Conversely, any splitting of an eigenform has isogenous summands. Here (E1,ω1) and (E2,ω2) in Ω 1 are isogenous if there is a surjective holo- M ∗ morphic map p : E1 E2 such that p (ω2)= tω1 for some t R. Connected sums also→ allow one to relate orbit closures in gen∈ us two to those in genus one. We conclude by sketching their use in the proof of Theorem 1.2. 1. Let Z = SL2(R) (X,ω) be the closure of an orbit in Ω1 2. Choose a splitting · M (X,ω) = (E1,ω1)#(E2,ω2), (1.1) I and let NI SL2(R) be the stabilizer of I. Then by SL2(R)-invariance, Z also contains⊂ the connected sums (n (E1,ω1))#(n (E2,ω2)) · I · for all n N . ∈ I 4 2. Let N G = SL (R) be the parabolic subgroup of upper-triangular ⊂ 2 matrices, let Γ = SL2(Z), and let N∆ and G∆ be copies of N and G diagonally embedded in G G. For u R we also consider the twisted diagonals × ∈ Gu = (g,n gn−1) : g G G G, { u u ∈ }⊂ × where n = ( 1 u ) N. u 0 1 ∈ The orbit of a pair of forms of genus one under the action of NI is isomor- phic to the orbit of a point x (G G)/(Γ Γ) under the action of N∆. By the classification of unipotent∈ orbits× ( 2),× we have Nx = Hx where § H = N , G , Gu (u = 0), N N, N G, G N, or G G. ∆ ∆ 6 × × × × 3. For simplicity, assume ω has simple zeros. Then if H = N∆ and H = G∆, we can find another point (X′,ω′) Z for which H6 = G G,6 which implies Z =Ω ( 11). ∈ × 1M2 § 4. Otherwise, there are infinitely many splittings with H = N∆ or G∆. Z The case H = N∆ arises when NI SL(X,ω) ∼= . If this case occurs for two different splittings, then SL(X,ω∩ ) contains two independent parabolic elements, which implies (X,ω) is an eigenform ( 5). § Similarly, the case H = G∆ arises when (E1,ω1) and (E2,ω2) are isoge- nous. If this case occurs for two different splittings, then (X,ω) is an eigenform by Theorem 1.8. 5. Thus we may assume (X,ω) Ω1ED for some D.